10 percent. We can confirm this by noting that when we discount the cash flows at 10

percent, the present value of the bond is equal to its $1,000 face value:

$100 $100 $1,100

PV at 10% = + + = $1,000.00

(1.10) (1.10)2 (1.10)3

But if you have to buy the 3-year bond for $1,136.16, the yield to maturity is only 5

percent. At that discount rate, the bond™s present value equals its actual market price,

$1,136.16:

$100 $100 $1,100

PV at 5% = + + = $1,136.16

(1.05) (1.05)2 (1.05)3

Calculating Yield to Maturity for the Treasury Bond

EXAMPLE 3

We found the value of the 6 percent coupon Treasury bond by discounting at a 5.6 per-

cent interest rate. We could have phrased the question the other way around: If the price

of the bond is $1,010.77, what return do investors expect? We need to find the yield to

maturity, in other words, the discount rate r, that solves the following equation:

$60 $60 $1,060

Price = + + = $1,010.77

(1 + r) (1 + r) (1 + r)3

2

FINANCIAL CALCULATOR

Bond Valuation on a Financial Calculator

Your calculator should now display a value of

Earlier we saw that financial calculators can compute

“1,010.77. The minus sign reminds us that the initial

the present values of level annuities as well as the pres-

cash flow is negative: you have to pay to buy the bond.

ent values of one-time future cash flows. Coupon

You can also use the calculator to find the yield to

bonds present both of these characteristics: the

maturity of a bond. For example, if you buy this bond

coupon payments are level annuities and the final pay-

for $1,010.77, you should find that its yield to maturity

ment of par value is an additional one-time payment.

is 5.6 percent. Let™s check that this is so. You enter the

Thus for the coupon bond we looked at in Example 3,

PV as “1,010.77 because you buy the bond for this

you would treat the periodic payment as PMT = $60,

price. Thus to solve for the interest rate, use the follow-

the final or future one-time payment as FV = $1,000, the

ing key strokes:

number of periods as n = 3 years, and the interest rate

as the yield to maturity of the bond, i = 5.6 percent. You

Hewlett-Packard Sharp Texas Instruments

would thus compute the value of the bond using the fol-

HP-10B EL-733A BA II Plus

lowing sequence of key strokes. By the way, the order

in which the various inputs for the bond valuation prob- PMT PMT PMT

60 60 60

lem are entered does not matter. FV FV FV

1000 1000 1000

N n N

3 3 3

Hewlett-Packard Sharp Texas Instruments PV PV PV

“1010.77 “1010.77 “1010.77

HP-10B EL-733A BA II Plus I/YR COMP i CPT I/Y

PMT PMT PMT

60 60 60

Your calculator should now display 5.6 percent, the

FV FV FV

1000 1000 1000

yield to maturity of the bond.

N n N

3 3 3

I/YR i I/Y

5.6 5.6 5.6

PV COMP PV CPT PV

To find the yield to maturity, most people use a financial calculator. For our Trea-

sury bond you would enter a PV of $1,010.77.4 The bond provides a regular payment of

$60, entered as PMT = 60. The bond has a future value of $1,000, so FV = 1,000. The

bond life is 3 years, so n = 3. Now compute the interest rate, and you will find that the

SEE BOX

yield to maturity is 5.6 percent. The nearby box reviews the use of the financial calcu-

lator in bond valuation problems.

Example 3 illustrates that the yield to maturity depends on the coupon payments that

you receive each year ($60), the price of the bond ($1,010.77), and the final repayment

of face value ($1,000). Thus it is a measure of the total return on this bond, accounting

for both coupon income and price change, for someone who buys the bond today and

holds it until maturity. Bond investors often refer loosely to a bond™s “yield.” It™s a safe

bet that they are talking about its yield to maturity rather than its current yield.

The only general procedure for calculating yield to maturity is trial and error. You

guess at an interest rate and calculate the present value of the bond™s payments. If

the present value is greater than the actual price, your discount rate must have been too

low, so you try a higher interest rate (since a higher rate results in a lower PV). Con-

4 Actually,

on most calculators you would enter this as a negative number, “1,010.77, because the purchase of

the bond represents a cash outflow.

263

264 SECTION THREE

versely, if PV is less than price, you must reduce the interest rate. In fact, when you use

a financial calculator to compute yield to maturity, you will notice that it takes the cal-

culator a few moments to compute the interest rate. This is because it must perform a

series of trial-and-error calculations.

A 4-year maturity bond with a 14 percent coupon rate can be bought for $1,200. What

Self-Test 3

is the yield to maturity? You will need a bit of trial and error (or a financial calculator)

to answer this question.

Figure 3.4 is a graphical view of yield to maturity. It shows the present value of the

6 percent Treasury bond for different interest rates. The actual bond price, $1,010.77, is

marked on the vertical axis. A line is drawn from this price over to the present value

curve and then down to the interest rate, 5.6 percent. If we picked a higher or lower fig-

ure for the interest rate, then we would not obtain a bond price of $1,010.77. Thus we

know that the yield to maturity on the bond must be 5.6 percent.

Figure 3.4 also illustrates a fundamental relationship between interest rates and bond

prices:

When the interest rate rises, the present value of the payments to be received

by the bondholder falls, and bond prices fall. Conversely, declines in the

interest rate increase the present value of those payments and result in

higher prices.

A gentle warning! People sometimes confuse the interest rate”that is, the return

that investors currently require”with the interest, or coupon, payment on the bond. Al-

though interest rates change from day to day, the $60 coupon payments on our Treasury

bond are fixed when the bond is issued. Changes in interest rates affect the present

value of the coupon payments but not the payments themselves.

FIGURE 3.4

The value of the 6 percent Price

bond is lower at higher $1,200

discount rates. The yield to

$1,150

maturity is the discount rate

at which price equals present $1,100

value of cash flows. $1,050

Bond price

Price = $1,010.77

$1,000

$950

$900

$850 Yield to maturity = 5.6%

$800

8%

0 2% 4% 6% 10% 12%

Interest rate

Valuing Bonds 265

RATE OF RETURN

When you invest in a bond, you receive a regular coupon payment. As bond prices

change, you may also make a capital gain or loss. For example, suppose you buy the 6

percent Treasury bond today for a price of $1,010.77 and sell it next year at a price of

$1,020. The return on your investment is the $60 coupon payment plus the price change

of ($1,020 “ $1,010.77) = $9.33. The rate of return on your investment of $1,010.77 is

RATE OF RETURN

Total income per period per

coupon income + price change

Rate of return =

dollar invested.

investment

$60 + $9.33

= = .0686, or 6.86%

$1,010.77

Because bond prices fall when market interest rates rise and rise when market rates

fall, the rate of return that you earn on a bond also will fluctuate with market interest

rates. This is why we say bonds are subject to interest rate risk.

Do not confuse the bond™s rate of return over a particular investment period with its

yield to maturity. The yield to maturity is defined as the discount rate that equates the

bond™s price to the present value of all its promised cash flows. It is a measure of the

average rate of return you will earn over the bond™s life if you hold it to maturity. In con-

trast, the rate of return can be calculated for any particular holding period and is based

on the actual income and the capital gain or loss on the bond over that period. The dif-

ference between yield to maturity and rate of return for a particular period is empha-

sized in the following example.

Rate of Return versus Yield to Maturity

EXAMPLE 4

Our 6 percent coupon bond with maturity 2002 currently has 3 years left until maturity

and sells today for $1,010.77. Its yield to maturity is 5.6 percent. Suppose that by the

end of the year, interest rates have fallen and the bond™s yield to maturity is now only 4

percent. What will be the bond™s rate of return?

At the end of the year, the bond will have only 2 years to maturity. If investors then

demand an interest rate of 4 percent, the value of the bond will be

$60 $1,060

PV at 4% = + = $1,037.72

(1.04) (1.04)2

You invested $1,010.77. At the end of the year you receive a coupon payment of $60

and have a bond worth $1,037.72. Your rate of return is therefore

$60 + ($1,037.72 “ $1,010.77)

Rate of return = = .0860, or 8.60%

$1,010.77

The yield to maturity at the start of the year was 5.6 percent. However, because interest

rates fell during the year, the bond price rose and this increased the rate of return.

Suppose that the bond™s yield to maturity had risen to 7 percent during the year. Show

Self-Test 4

that its rate of return would have been less than the yield to maturity.

Is there any connection between yield to maturity and the rate of return during a par-

ticular period? Yes: If the bond™s yield to maturity remains unchanged during an invest-

266 SECTION THREE

ment period, its rate of return will equal that yield. We can check this by assuming that

the yield on 6 percent Treasury bonds stays at 5.6 percent. If investors still demand an

interest rate of 5.6 percent at the end of the year, the value of the bond will be

$60 $1,060

PV = + = $1,007.37

(1.056) (1.056)2

At the end of the year you receive a coupon payment of $60 and have a bond worth

$1,007.37, slightly less than you paid for it. Your total profit is $60 + ($1,007.37 “

$1,010.77) = $56.60. The return on your investment is therefore $56.60/$1,010.77 =

.056, or 5.6 percent, just equal to the yield to maturity.

When interest rates do not change, the bond price changes with time so that

the total return on the bond is equal to the yield to maturity. If the bond™s

yield to maturity increases, the rate of return during the period will be less

than that yield. If the yield decreases, the rate of return will be greater than

the yield.

Suppose you buy the bond next year for $1,007.37, and hold it for yet another year, so

Self-Test 5

that at the end of that time it has only 1 year to maturity. Show that if the bond™s yield

to maturity is still 5.6 percent, your rate of return also will be 5.6 percent and the bond

price will be $1,003.79.

The solid curve in Figure 3.5 plots the price of a 30-year maturity, 6 percent Trea-

sury bond over time assuming that its yield to maturity remains at 5.6 percent. The price

declines gradually until the maturity date, when it finally reaches face value. In each

period, the price decline offsets the coupon income by just enough to reduce total return

to 5.6 percent. The dashed curve in Figure 3.5 shows the corresponding price path for

a low-coupon bond currently selling at a discount to face value. In this case, the coupon

income would provide less than a competitive rate of return, so the bond sells below par.

Its price gradually approaches face value, however, and the price gain each year brings

its total return up to the market interest rate.

FIGURE 3.5

Bond prices over time, $1,100 Price path for bond currently at

assuming an unchanged yield a premium over face value

to maturity. Prices of both $1,050

premium and discount bonds

approach face value as their $1,000

Bond price